## Laplaza sets, or . . . for pseudo algebras (2008)

### BibTeX

@MISC{Fiore08laplazasets,,

author = {Thomas M. Fiore and Po Hu and Igor Kriz},

title = {Laplaza sets, or . . . for pseudo algebras},

year = {2008}

}

### OpenURL

### Abstract

### Citations

251 |
Braided tensor categories
- Joyal, Street
- 1993
(Show Context)
Citation Context ...gram in [12], which requires that the left and right unit coherence isomorphisms agree on the unit, is not included in our coherence diagrams, but this diagram was proved to follow from the others in =-=[8]-=-). The equivalence of the category of unbiased symmetric monoidal categories with the category of symmetric monoidal categories is essentially the coherence theorem of Mac Lane. Let C be an operad. Th... |

82 |
Categories for the Working
- Lane
- 1971
(Show Context)
Citation Context ...rd a can be turned into another word b in two different ways, using a different sequence of identities. To such a situation, there corresponds in an obvious way a coherence diagram (see, for example, =-=[12]-=-). It was suggested in [5] that all such diagrams should be required to commute in a pseudo algebra. It turns out, however, that such a requirement is unreasonably strong. For example, if the algebras... |

77 |
Two dimensional monad theory
- Kelly, Blackwell, et al.
- 1989
(Show Context)
Citation Context ...22 1. Introduction Generalizing algebras to pseudo algebras is a basic idea which has recently become important in axiomatization of conformal field theory [2,5–7], as well as in other subjects, e.g. =-=[1]-=-. While the exact settings vary, the kind of algebras we are using generally have a set of operations and a set of identities (equations) the operations are required to satisfy. The corresponding noti... |

74 |
Higher operads, higher categories
- Leinster
- 2004
(Show Context)
Citation Context ...ras modelled on one set (“1-sorted algebras”) involves the language of operads, and its interplay with the language of theories. This idea is also basically due to M. Gould. See [3] and [4], and also =-=[11]-=- for background. 1 The relevant concepts are defined in the next section. The foundational results of [2] remain valid and can be generalized to the new context, and hence all the substantive results ... |

47 |
The definition of conformal field theory, Topology, Geometry and quantum field theory
- Segal
(Show Context)
Citation Context ...n of 2-theory is defined in [5]. The main example of interest here is the 2-theory of commutative monoids with cancellation, which we use to make the symmetric definition of conformal field theory in =-=[13]-=- more precise. We recapitulate these definitions here before turning to Laplaza sets for 2-theories. Definition 3.1. A 2-theory consists of a natural number k, a theory T ,sets Θ(w; w1,...,wn) for all... |

38 | Conformal field theory and elliptic cohomology
- Hu, Kriz
(Show Context)
Citation Context ...omorphisms. Such diagrams are generally known as coherence diagrams. The question of what coherence diagrams one should select is trickier than it may appear, and is the main subject of this note. In =-=[5]-=-, the following scheme was suggested for selecting coherence diagrams: take all diagrams which can be “reasonably expected” to commute. This means, take any word in our algebra which can be formed by ... |

29 |
Lawvere, Functorial semantics of algebraic theories
- William
(Show Context)
Citation Context ... by Laplaza’s diagrams, we call Laplaza sets in recognition of his contribution. 2. Pseudo algebras with Laplaza sets in theories Let us recall here the notion of a theory, which was first defined in =-=[10]-=-. We will stick to the “universal algebra” point of view, which is more advantageous for defining pseudo algebras. Definition 2.1. Let Γ be the category with objects 0, 1, 2,... where 0 =∅and k := {1,... |

26 | Closed and open conformal field theories and their anomalies - Hu, Kriz |

26 |
Coherence for distributivity. Coherence in categories
- Laplaza
- 1972
(Show Context)
Citation Context ... on one side. In this case of pseudo commutative semi-rings, the pseudo algebras should be symmetric bimonoidal categories. The correct condition limiting coherence diagrams was discovered by Laplaza =-=[9]-=-. The condition is somewhat technical and will be explained in Example 2.7 below. It is not obvious what general scheme would select coherence diagrams “correctly” in accordance with what one expects ... |

20 | Pseudo limits, biadjoints, and pseudo algebras: Categorical foundations of conformal field theory
- Fiore
(Show Context)
Citation Context ... language of theories. This idea is also basically due to M. Gould. See [3] and [4], and also [11] for background. 1 The relevant concepts are defined in the next section. The foundational results of =-=[2]-=- remain valid and can be generalized to the new context, and hence all the substantive results of [5–7] remain in effect. One kind of algebra which is of interest in conformal field theory however is ... |

3 | 2008), Coherence for categorified operadic theories
- Gould
(Show Context)
Citation Context ...of C ′ which map to the same element of C.Apseudo C-algebra is a C ′ -algebra, i.e. a morphism C ′ → EndCat(X) of categorical operads. This is the way that pseudo algebras over operads are defined in =-=[3]-=- and [4]. Proposition 2.6. Let T be a theory which is free on an operad C, and let S be the corresponding Laplaza set. Then a pseudo (T , S)-algebra is the same thing as a pseudo C-algebra. Proof. Let... |

3 | The categorification of a symmetric operad is independent of signature
- Gould
- 2007
(Show Context)
Citation Context ...g as symmetric monoidal categories, and those will not in general be equivalent to categories satisfying (2) (see Proposition 2.5 below). The authors thank M. Gould for this example, see Section 6 of =-=[4]-=-. To correct this, one must generalize the notion of pseudo algebra in a way that allows us to limit the scope of coherence diagrams required, so that “bad diagrams” such as the one mentioned above ca... |

2 | On modular functors and the ideal Teichmüller tower - Hu, Kriz |